Abstract
We show that the mean-variance preferences have counterfactual implications for a risk averse long term decision maker. In the simple case of dynamic portfolio choice, we show that the optimal certainty equivalent is decreasing with the investor’s horizon towards its lower bound, the riskless rate. For some horizons (less than 25 years in our simulations), the economic value of diversification is 0 and therefore the optimal portfolio strategy is a buy and hold one in the riskless asset. Therefore, under-diversification is optimal. These results question the usefulness of the mean variance setting for long term dynamic decision making.
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- 1.
See Ingersoll (1987) and Cochrane (2005) for a textbook treatment of the Finance applications. See also Choi (2008) for a literature review of its applications in Supply Chain Contracts. CARA utility combined with normal distributions leads to mean variance criterion and this is the work horse approach for the economic theory of information transmission by prices; see Marin and Rahi (2000) for a typical example.
- 2.
See the recent extension by Zhang et al. (2009) and the references therein.
- 3.
See Amenc et al. (2011).
- 4.
See, for example, Jondeau and Rockinger (2012).
- 5.
See Cochrane (2005), chapter 1 and 4 for details.
- 6.
See Lioui (2013).
- 7.
In an equilibrium setting, recall that the pricing kernel is the marginal utility of consumption of the representative investor. See Cochrane (2005) for details.
- 8.
In a dynamic setting, the portfolio composition has two components, a mean-variance component and an intertemporal hedging component. The first one is always decreasing with risk aversion while the behavior of the second one is ambiguous. Nevertheless, since this component is mainly related to the volatility of the predictors that make the opportunity set time varying, it is in general low in size relative to the mean variance component, although economically sizeable.
- 9.
This can be inferred from Eq. (10) of the relative risk aversion.
- 10.
Recall that when markets are complete, the number of risky assets must be equal to the number of sources of risk.
References
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Lioui, A., 2013, Time consistent vs. time inconsistent dynamic asset allocation: Some utility cost calculations for mean variance preferences, Journal of Economic Dynamics and Control, 37(5), 1066 -1096. http://app.edhec.edu/faculty/spring/accueilFlow?execution=e2s1
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Appendix
Appendix
Proof of Proposition 1:
Investing in the risky assets is optimal if the following condition hold:
As a consequence, we have:
On the other hand, we have:
A sufficient condition for this derivative to be negative is:
Since we have:
Condition (A5) holds whenever:
Using (A2), condition (A6) becomes:
Condition (A7) holds when
holds and this will always be the case since the right hand side is always negative while the left hand side is always positive.
Proof of Proposition 2:
The first order condition writes:
where \(\uptheta \) is the Lagrange multiplier. Using the budget constraint, the Lagrange multiplier is given by:
This could be written as:
It has a similar interpretation as the one obtained in (4). It is positive when the present value of all maximal consumptions is greater than initial wealth. The Dollar certainty equivalent for consumption at time t is such that:
Therefore, using the expression for the optimal consumption, one has:
The certainty equivalent associated with the optimal consumption plan will thus be:
and therefore:
The annualized certainty equivalent becomes:
Substituting for the Lagrange multiplier yields:
Under the specific setting outlined above, this becomes:
Note also that in the particular case of constant opportunity set, the condition (A11) implies:
The maximum certainty equivalent is the one that would obtain when consumption is equal to the upper bound at each time t in the future. Therefore:
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Lioui, A. (2014). A Paradox of the Mean Variance Setting for the Long Term Investor. In: El Ouardighi, F., Kogan, K. (eds) Models and Methods in Economics and Management Science. International Series in Operations Research & Management Science, vol 198. Springer, Cham. https://doi.org/10.1007/978-3-319-00669-7_5
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DOI: https://doi.org/10.1007/978-3-319-00669-7_5
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